Reasoning Shortcuts and Value Symmetries: What Symmetry Permits, Architecture Realizes, and Optimization Selects

Reasoning shortcuts are solutions of a neurosymbolic system's rules that produce correct predictions through unintended concepts. A recent framework of Takemura, Inoue, and Nishino analyzes them through an automorphism group of value relabelings and asks, as its central open question, when rules pin concepts down. We first show that the framework's key definition, one shared permutation applied at every position, does not apply as stated to any of the four heterogeneous benchmarks it was evaluated on, and that the most direct embedding, padding domains to a common size, produces confident false pathology: 90.91% of solution pairs reported unexplained on CLE4EVR, where every well-defined member of the hierarchy we introduce reports 0%, and the padded verdict's content rotates with configuration-file ordering. Re-measuring eleven rule families under fifteen pre-specified predictions (thirteen confirmed), unexplained-pair rates span 0% to 99.9999% and track provable structure: six theorems give sufficient conditions for transitivity and its failure, including a Free Slot Lemma certifying Kandinsky's pathology from syntax alone. For circuit-given rules, deciding symmetry-inertness of a coordinate is coNP-complete; nontrivial-automorphism existence is coNP-hard under randomized reductions, lies in $Σ_2^p$, is not $Σ_2^p$-complete unless PH collapses, and on monotone circuits is coNP-complete outright. In the Boolean case transitivity is classified exactly: automorphisms explain everything iff the solution set is an affine coset. Weakly supervised models place all 94 observed shortcuts at the one level the componentwise theory flags and none at the 48 it certifies transitive; twelve typed-ambiguous levels produce none, separating what symmetry permits from what optimization selects, and a dual-head control replicates the geography. All numbers trace to released artifacts.